| L(s) = 1 | − 16·2-s + 64·4-s + 266·7-s + 1.02e3·8-s − 582·9-s + 1.74e3·11-s − 4.25e3·14-s − 1.63e4·16-s + 9.31e3·18-s − 2.79e4·22-s + 9.47e3·23-s + 2.92e4·25-s + 1.70e4·28-s + 2.22e4·29-s + 6.55e4·32-s − 3.72e4·36-s + 6.00e3·37-s + 6.28e4·43-s + 1.11e5·44-s − 1.51e5·46-s − 4.68e4·49-s − 4.67e5·50-s − 1.52e5·53-s + 2.72e5·56-s − 3.56e5·58-s − 1.54e5·63-s + 7.86e5·64-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 0.775·7-s + 2·8-s − 0.798·9-s + 1.31·11-s − 1.55·14-s − 4·16-s + 1.59·18-s − 2.62·22-s + 0.778·23-s + 1.86·25-s + 0.775·28-s + 0.914·29-s + 2·32-s − 0.798·36-s + 0.118·37-s + 0.790·43-s + 1.31·44-s − 1.55·46-s − 0.398·49-s − 3.73·50-s − 1.02·53-s + 1.55·56-s − 1.82·58-s − 0.619·63-s + 3·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4460537853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4460537853\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - 38 p T + p^{6} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 194 p T^{2} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 5842 p T^{2} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 874 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4755578 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 748834 p T^{2} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 84379322 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 206 p T + p^{6} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 11146 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1020892802 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3002 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6189133442 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 31418 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 16309886018 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 76406 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 71539567322 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 27356175482 T^{2} + p^{12} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 495242 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 184406 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 298950552578 T^{2} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 534934 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 142873131578 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 597656180162 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1002631840898 T^{2} + p^{12} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.56593118546249998153867639690, −20.33687755471626900603484997699, −19.97041013568860581398130146807, −19.11798010630834252916645370873, −18.67158597206423904655628524524, −17.82701406890360998772407454060, −17.19068626466928186347118288612, −17.00653500585425297663577113249, −16.03839533185168513375849669959, −14.42419963223029106500639446527, −14.20134740872490013054366128409, −12.86581600670512219937921367609, −11.40316797027950834772575422884, −10.75923984241788276839323529391, −9.625020434465131307411498781647, −8.798957585047171515468592210723, −8.311442177711380676451943091851, −7.01191900691460940820953897559, −4.71547017025821628032419045203, −1.07733282911391212732378645122,
1.07733282911391212732378645122, 4.71547017025821628032419045203, 7.01191900691460940820953897559, 8.311442177711380676451943091851, 8.798957585047171515468592210723, 9.625020434465131307411498781647, 10.75923984241788276839323529391, 11.40316797027950834772575422884, 12.86581600670512219937921367609, 14.20134740872490013054366128409, 14.42419963223029106500639446527, 16.03839533185168513375849669959, 17.00653500585425297663577113249, 17.19068626466928186347118288612, 17.82701406890360998772407454060, 18.67158597206423904655628524524, 19.11798010630834252916645370873, 19.97041013568860581398130146807, 20.33687755471626900603484997699, 21.56593118546249998153867639690
